Asymptotic evaluation of certain markov process expectations for large time. IV

Abstract

In this paper we extend our asymptotic results [l], [2] to a more general setting. Let R be a space of functions w (. ) on -00 < t < 00 with values in a Polish space X. We assume R consists of functions with discontinuities only of the first kind, normalized to be right continuous, and with convergence induced by the Skorohod topology on bounded intervals. In this case, R is itself a Polish space. Denote by 0: the corresponding space of functions on [t, 00) with values in X.We denote by S S the a-field in R generated by w (a) for s 5 a 5 t. We denote the translation map on R by Or, i.e., (O,w)(s) = w (s + t).Let Po,, be a Markov process on a,' starting from x E X satisfying the hypothesis that the mapping x +PO,, is weakly continuous (which implies the Feller property for the process PO,,).Let w E R and, for each t >0, define wt by w r ( s ) = w ( s ) , O S s < t ,wt(s + t ) = w f ( s ) for all s E (-00, 00). Now, (6,wt)(T) = wt(s + T ) , 0 5 s 5 t, and for any set A c R we defineIf we let &(R) denote the space of stationary processes on R, we see that, for each w E R and each t > 0, I?',,( a ) E &(R). This is because, for any a > 0,

Large-time option pricing using the Donsker–Varadhan LDP—correlated stochastic volatility with stochastic interest rates and jumps

“…In [Var84] (page 34) and [DV76], [DV83] it is shown that the following five conditions imply the LDP upper bound in (2.2):…”

“…These two conditions are given in [Var84], [DV83], where the LDP is proved as a corollary of a more general LDP on path space in terms of the entropy function (see Theorem 13.1.31 in [Var84]). These two conditions simplify the more cumbersome conditions for the LDP lower bound given on page 393 in [DV76].…”

Large deviations for two scaled diffusions

“…Note at first LDP for family µ ε (dz) = ν ε ([0, 1], dz) (on the space of probability measures supplied by Levy-Prohorov's metric) proved by Donsker and Varadhan [6], [7], [8], [9] for a wide class of Markov processes ξ ε t = ξ t/ε . Corresponding rate function obeys an invariant form: for any probabilistic measure µ on R…”

“…In contrast with Freidlin and Wentzell [12], Donsker and Varadhan [6] - [9], Gärtner [13], and Veretennikov [26] - [27], and many others (see e.g. Acosta [1], Dupuis and Elis [5]) our method of proof is based on Puhalskii's theorem [19] - [20] and relies concepts of exponential tightness and LD relative compactness.…”

Bernstein Processes Associated with a Markov Process

“…By Proposition 3.7, there exists Q satisfying Q t = µ t for t = 0, 1 and A + (Q) < +∞. By Lemma 3.4 (3.11b), we have, for any such Q, 19) which is finite. Since the last term in (3.19) is independent on Q (i.e., depends only on (µ 0 , µ 1 )), this result follows from Proposition 3.7.…”